cp's OEIS Frontend

a(n) is the number of triangles with integer sides and perimeter n.

Also a(n) is the number of triangles with distinct integer sides and perimeter n+6, i.e., number of triples (a, b, c) such that 1 < a < b < c < a+b, a+b+c = n+6. - Roger Cuculière

With a different offset (i.e., without the three leading zeros, as in A266755), the number of ways in which n empty casks, n casks half-full of wine and n full casks can be distributed to 3 persons in such a way that each one gets the same number of casks and the same amount of wine [Alcuin]. E.g., for n=2 one can give 2 people one full and one empty and the 3rd gets two half-full. (Comment corrected by Franklin T. Adams-Watters, Oct 23 2006)

For m >= 2, the sequence {a(n) mod m} is periodic with period 12*m. - Martin J. Erickson (erickson(AT)truman.edu), Jun 06 2008

Number of partitions of n into parts 2, 3, and 4, with at least one part 3. - Joerg Arndt, Feb 03 2013

For several values of p and q the sequence (A005044(n+p) - A005044(n-q)) leads to known sequences, see the crossrefs. - Johannes W. Meijer, Oct 12 2013

For n>=3, number of partitions of n-3 into parts 2, 3, and 4. - David Neil McGrath, Aug 30 2014

Also, a(n) is the number of partitions mu of n of length 3 such that mu_1-mu_2 is even and mu_2-mu_3 is even (see below example). - John M. Campbell, Jan 29 2016

For n > 1, number of triangles with odd side lengths and perimeter 2*n-3. - Wesley Ivan Hurt, May 13 2019

Number of partitions of n+1 into 4 parts whose largest two parts are equal. - Wesley Ivan Hurt, Jan 06 2021

For n>=3, number of weak partitions of n-3 (that is, allowing parts of size 0) into three parts with no part exceeding (n-3)/2. Also, number of weak partitions of n-3 into three parts, all of the same parity as n-3. - Kevin Long, Feb 20 2021

Also, a(n) is the number of incongruent acute triangles formed from the vertices of a regular n-gon. - Frank M Jackson, Nov 04 2022

Rows alternate between all 1's and alternating 1's and 0's. A 'mixed' sequence array: rows alternate between the rows of the sequence array for the all 1's sequence and the sequence array for the sequence 1,0,1,0,...

Column 2*k has g.f. x^(2*k)/(1-x); column 2*k+1 has g.f. x^(2*k+1)/(1-x^2).

Row sums are A029578(n+2). Antidiagonal sums are A106466.

This triangle is the Kronecker product of an infinite lower triangular matrix filled with 1's with a 2 X 2 lower triangular matrix of 1's. - Christopher Cormier, Sep 24 2017

From Peter Bala, Aug 21 2021: (Start)

Using the notation of Davenport et al.:

This is the double Riordan array ( 1/(1 - x); x/(1 + x), x*(1 + x) ).

The inverse array equals ( (1 - x)*(1 - x^2); x*(1 - x), x*(1 + x) ).

They are examples of double Riordan arrays of the form (g(x); x*f_1(x), x*f_2(x)), where f_1(x)*f_2(x) = 1. Arrays of this type form a group under matrix multiplication. For the group law see the Bala link. (End)

The old definition was "Lexicographically earliest sequence starting with a(1) = 0 such that a(n) is the number of pairs of contiguous terms whose sum is a(n)".

From Paul Curtz, Apr 27 2019: This can be written as a triangle:

0

1 1

1 2 1

2 2 2 2

2 3 2 3 2

3 3 3 3 3 3

3 4 3 4 3 4 3

...

We build up a planar graph with hexagonal cells, based on the square grid. There are six kinds of edges.

A U edge is drawn from (x,y) to (x,y+1);

a U^{-1} edge is drawn from (x,y) to (x,y-1);

an L edge is drawn from (x,y) to (x-1,y+1);

an L^{-1} edge is drawn from (x,y) to (x+1,y-1);

an R edge is drawn from (x,y) to (x+1,y+1); and

an R^{-1} edge is drawn from (x,y) to (x-1,y-1).

The construction starts in generation 0 with a single node at the origin (see illustration). At generation 1 we draw a U line from the origin to (0,1).

The graph is then extended using the following rules.

Every U is followed by a pair of lines, L and R;

every L is followed by a U;

every L^{-1} is followed by a pair U^{-1} and R; and

every R is followed by a pair U and L^{-1}.

Lines that fall outside the first quadrant are ignored, and lines that would coincide with existing lines are ignored.

Lines of type U^{-1} and R^{-1} do not need to be followed by anything.

The node numbers in the illustration indicate at which generation the node is reached. This is also the graph distance from the origin.

The number of nodes that are added at the n-th generation, for n >= 0, is given by 1, 1, 1, 2, 4, 4, 5, 5, 7, 7, 8, 8, 10, 10, 11, 11, 13, 13, 14, 14, 16, 16, 17, 17, 19, 19, ..., with G.f. = (-x^7+x^6+x^4+x^3+1)/((1-x)*(1-x^4)). This is essentially A265428.

The total number of nodes after the n-th generation, for n >= 0, is 1, 2, 3, 5, 9, 13, 18, 23, 30, 37, 45, 53, 63, 73, 84, 95, 108, 121, 135, 149, 165, 181, 198, ... This is essentially A265429.

The number of hexagons that are added at the n-th generation, for n >= 0, is given by 0, 0, 0, 0, 0, 1, 1, 2, 1, 3, 2, 4, 2, 5, 3, 6, 3, 7, 4, 8, 4, 9, 5, 10, 5, 11, 6, 12, 6, 13, ..., with G.f. = x^5*(1+x+x^2)/((1-x^2)*(1-x^4)). This is essentially A106466.

Consider the triangle T(n, k) = A003989(n, k) = gcd(n-k+1, k), n >= 1, k = 1..n. Then a(n) = Sum_{k=0..floor(n/2)} T(n-k+1, k+1), for n >= 0. - R. J. Mathar, May 11 2018 [adjusted to the definition of A003989. - Wolfdieter Lang, May 12 2018]